Lefschetz Pencils on surfaces and an application to Seshadri constants

@misc{hanumanthu2026lefschetz,
  title = {{Lefschetz Pencils on surfaces and an application to Seshadri constants}},
  author = {Krishna Hanumanthu and A. J. Parameswaran},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23485},
  eprint = {2610.23485},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23485},
}

TY  - UNPB
TI  - Lefschetz Pencils on surfaces and an application to Seshadri constants
AU  - Krishna Hanumanthu
AU  - A. J. Parameswaran
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23485
UR  - https://arxiv.org/abs/2610.23485
AB  - Let X be a nonsingular complex projective surface and let L be a very ample line bundle on X. We study the question: when does a general pencil in the linear system |L| consist of only irreducible and reduced curves? We give a precise condition which guarantees an affirmative answer: in fact, this holds if and only if the genus of a general member of |L| exceeds the irregularity of X. The proof is topological, using vanishing cycles and the theory of Lefschetz pencils. As an application, combined with a construction due to Miranda, we prove that for any positive rational number δ there exist a blow up Y → X at finitely many points, an ample line bundle M on Y and a point x ∈ Y such that ε(Y,M,x) = δ; Miranda proved this when X = ℙ^2.
ER  -